# What is the iff condition for a preference with linear Engel curves (all Engal curves are linear)?

If we restrict the consumption at $$\mathbb R_+^n$$, then it seems like we are implicitly assuming that the Engel curve pass through the origin, so the iff condition would be homothetic preference.

However, if we let $$x\in\mathbb R^n$$, other classes of preference would also give linear Engel curves. For example, those represented by a quasi linear utility in two dimensions: $$u(x)=x_1+f(x_2)$$; in this case, the Engel curves are parallel lines.

It could be assumed that, as in the standard consumer theory, the preference is non-satiated and continuous, so the indifference sets are curves.

Are negative incomes (perhaps even negative prices) also allowed? Unless they are, non-horizontal or non-vertical lines are impossible as Engel curves because they have to intersect the negative quadrant of the coordinate system. I will come back to this later.

Without additional restrictions (perhaps a lot of additional restrictions) on the preference relations, for any non-negative sloped line I can fabricate a utility function such that the line will be its Engel curve.

Let the line $$L$$ consist of the points $$((x,y)\in\mathbb{R}^2|c = a \cdot x + b \cdot y)$$ where $$a,b,c \in \mathbb{R}$$. I will denote this set of points by $$L$$, the line. It is assumed that we are talking about an actual line, so $$a,b$$ are not simultaneously equal to $$0$$.

Let $$U(x,y) = \left\{\begin{array}{lc} \arctan (b \cdot x + a \cdot y) & \text{ if } (x,y) \in L \\ -\pi & \text{ if } (x,y) \notin L. \end{array} \right.$$

Claim The utility function will always be maximized by a point on $$L$$.

This is easy, because $$\arctan$$ maps to $$(-\frac{\pi}{2},\frac{\pi}{2})$$, thus its points always yield greater utility than other points. We specified that the slope of $$L$$ is non-negative, so for any positive price pair the budget line will cross it, enabling us to chose a point on $$L$$.

We now come back to the question of whether income $$I$$ can be negative. If they can be, than every point $$L$$ will be a solution of the consumers utility maximization problem for a certain $$I,p_x,p_y$$. The reason for this is that $$b \cdot x + a \cdot y$$ creates an ordering of the points of $$L$$. (Lines perpendicular to $$L$$ have a slope of $$-b/a$$, these are the level curves of $$b \cdot x + a \cdot y$$.) Select a point $$(x_0,y_0) \in L$$. By setting $$I$$ to $$I = p_x \cdot x_0 + p_y \cdot y_0,$$ the consumer has barely enough money to purchase this basket. If $$L$$ is non-negative sloped and prices are positive then no points with higher utility are attainable.

The construction I gave is not unique. One could also use the continuous utility function $$\hat{U}(x,y) = \min\left(a \cdot \left(x - \frac{c}{a} \right); - b \cdot y\right)$$ to achieve the same result. (If $$a = 0$$ one may use $$\hat{U}(x,y) = \min\left(a \cdot x; - b \cdot \left(y - \frac{c}{b} \right)\right)$$ instead.)

• Thanks for your help answer! I might start another question with additional restrictions. What are some of the reasonable yet not-too-strong assumptions? – High GPA Jun 9 '19 at 15:36
• @HighGPA I have no clue. I don't think this is a much studied problem, the Engel-curve seems to be an intro to Micro tool, and in that context people usually study non-negative good baskets. – Giskard Jun 9 '19 at 15:41